Hello, 
This looks to be spreading dogbane, which is a species of Indian hemp.  (Apocynum androsaemifolium).  Here is some more information about this native plant: 
https://www.fs.usda.gov/wildflowers/plant-of-the-week/apocynum_androsaemifolium.shtml
Please note that the milky sap is toxic to humans and animals, so take care when handling this plant.  
As far as transplanting goes, you can transplant perennials anytime until the ground freezes in the fall, or wait to transplant them in the spring. Fall is an excellent time to transplant perennials because your plants will then have three seasons to establish a good root system before hot summer weather sets in next year.  Another good option is to mark the plant so you can identify it after it has lost its leaves, and move it in the spring after the ground thaws but before growth starts.
I hope this is helpful.  Thank you for contacting Ask Extension.
Ruth